The email and the user for the support department are different. Email can only be imported to the department if the TO address matches that of the department email. Since you're authenticating with Google using a different address, I wonder if this might be part of the issue.
If the ticket import is stalling there may be other issues. Please ensure that the mailparse pecl extension is installed. Also, you could disable your cron job, wait 5 minutes, email the ticket address, and then run the cron manually under Settings > System > Automation. This won't work if the task is still hung (spinning icon under Settings > Company > Automation). This should show you the ticket cron output. If it seems to stall, or still doesn't work and the task shows that it is locked, try enabling error reporting and running the cron again (after at least 5 minutes).
To enable error reporting, edit /config/blesta.php and change
Configure::errorReporting(0);
to
Configure::errorReporting(-1);